4-

UNIVERSITY OF ILLINOIS LIBRARY

Ciass

Book

Volume

■4, ^ 4r ^

f 4

GRAPHIC EVALUATION OF TRIGONOMETRIC FUNCTIONS

OF COMPLEX VARIABLES

BY r^!

TRUMAN LEE KELLEY

THESIS

For the Degree of BACHELOR OF ARTS IN MATHEMATICS

COLLEGE OF SCIENCE OF THE

UNIVERSITY OF ILLINOIS June 1909

UNIVERSITY OF ILLINOIS

June 1 9

THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION BY

Mr. Triman Lee Kelley

ENTITLED QRAPniC EVM;XJATIOj\T OF TRIG-01Tpi£ETKIC IMC.TIOIS

OP COMPLEX VARIABLES

IS APPROVED BY ME AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE

DEGREE OF bachelor Of Arts

in Mathematics

APPROVED:

HEAD OF DEPARTMENT OF Mathematics

145140

UlUC

TABLE OF COHTEHTS

Pages

Art .1 Introduction 1-3

Graphic representation of functions of complex variables .

Art. 2 Properties of analytic functions 4

Art. 3 Discussion of W— sin z 5-7

Art. 4 Mapping of ¥— sin z 7 -11

(1) Pundainental region 7

(2) Relation between the four quadrants 9

(3) Mapping of lines parallel to the axes 10 Art. 5 Explanation of the maps 11-16

(1) General description 11

(2) Map I 12

(3) Map II 13

(4) Map III . 13

(5) Description of the ellipsograph 14

(6) Construction of maps 15 Art. 6 Evaluation of sin(a4-ib), where b>5.5. 16-18 Art. 7 Evaluation of sin(a-hib), where -^)a.yOr)^ 18-20 Art .8 Accuracy/- of the maps 20-21 Art .9 Examples 21-25

Tables of multiples of TTand logglO 25 Art .10 Evaluation of cos z, tan z, cot z, sec z,

CSC z 26

Bibliography 27 Maps I, II and III.

II

Art. 1.- lUTROLUCTIOX .

The method of the analytic geometry of representing, graphically, numerical values, by distances on a straight line is here assuiaed. Moreover, v/e assume the one to one correspondence of the real numbers in a continuum and the points of a straight line .

Y/hen numbers of the type \]~-B, called imaginaries, and represented by \p7" (b = \rB-) , or, simply ib, were introduced, a method was devised for representing then graphically. The common representation of such a number is by measurement of the distance b a.long a line at right angles to the one used for representing rea,l numbers, and passing through the same origin. ib is repre- sented by a point at distance b, measured upward from th:; origin and -ib by a point at distr^nce b, belov/ the origin. An extension of the preceding is the representation of a number of the type, (a -I- ib), by measuring a distance a along the line chosen to re- present real numbers, called the axis of reals, and from this point a distance bi along a line parallel to the line representing imaginary numbers, called the axis of imaginaries. By this method all real numbers (± a), all imaginary numbers (iib), and

all complex numbers (ictiib), may be FIGURE represented upon a single plane. The f~

acGompanyinff diagram illustrates this q

method of representation. ^ -

^1

^(--^]

This representation upon the complex plane may be used to represent functions of complex numbers. Por convenience tv/o

2

5

such planes are used, one to represent the complex variable and the other the function. V/e shall call the former the Z-plane and the latter the ¥-plane . The usual method of representing the functional relation betv/een? z and W is, ¥ = f (2) . It is under- stood th£it W ajid z are, in general, both comijlex nutnbers . We may also v/rite, 2; r x + iy, and, W : IJ 4- iV, in v/hich case U and V are both functions of x,y.

If the independent varia,ble 2, in the equation W z f(z), is allowe to take a succeBsion of values, FIGURE II

say those represented by points on the curve £, drawn in the Z-plane, then V/ taJces a sequence of values represented, let us assume, by the line £, drawn in the W-plane . Py means of the relation, V/ = f(z), v/e say that the curve _c, of the Z-plane maps into the curve C_, of the W-pla.ne .

If 2 takes a series of values, represented by a line parallel to the axis of reals, or the axis of imaginaries, this will ffive some curve in the ¥-plane. If all the lines parallel to the axis of reals, and those parallel to 1 he axis of ima,ginar- ies, in the Z-plane, are thus mapped on the W-plane, the value of ¥ for any z, in W :: f(z), may be read by locating the point in the ¥-plane . The following example v^ill illustrate this:

X.

c

Given, ¥ = ^.

Let us d^-t ermine hat the lines parallel

to the X s.nd y axis, respectively, map into in the ¥-plane : 1 ^ X - iy X iy

U f iV=

X + ly

Equa.ting reals and imap:inar ies , we have,

TT ^

U — A ^-A

Eliminating y f^rom these two equations, v/e have.

This equation, for values x = a constant, is represented by a cir- cle passing thi'ough the origin, with center upon the U-axis and

at d. distance i_ from the origin. 2x

Similarly, w e hav e ,

which is represented by a circle passing through the origin,

with center upon the V-axis ana at a distance 2;— from the origin,

2y

for y r c, a constant. In the accompanying fio-urc, representing the map of W :: i, lines in the Z-plane map into the lines corres- pondingly lettered in the ¥/-plane.

W- Plane

III

I

y

c

•L,

— —

4

Art. 2.- PROPERTIES OF A:IALYTIC RUIJCTIOI^S .

The follov/inn- propositions, found in any text book deal- ing with, functions of complex variables, are assumed:

(1) The necessary and sufficient condition that V,' (V/ : UfiV) is an analytic function of z [z z x+iy) is that the following partial dif f ere-nt ial equations are satisfied:

ax - 3y' ay - ax

(2) If W is an analytic function of z, then U =l|^(x,y),

and V =0(x,y). U and V are conjugate functions of x,y and the tv/o systems of curves are orthogonal.

(3) If W = f(z) is a.n analytic function, the mapping of the Z-plane into the W-plane is a conformal representation, i.e., infinitesimal elements are preserved.

(4) z is a sinp-le-valued and continuous function of V/ if for every continuous series of points in the Z-plane there is a con- tinuous series of points in the V/- plane.

(5) If z is a single-valued and continuous function of W, and W a single valued function of z, th- n there exist fundajnental regions in the z-plane, such that each maj) once and only once into the entire W-plane. In this case, l=f(z), is a periodic function and the limits of the fundamental region are determined by the period of f (z ) .

(6) The trigonometric functions of complex numbers are de- fined as follows:

1

cot z r: sec z n

CSC

tan z

1_^^

C 0 s~ z"

1

COS z sm z

Art. 3.- DISCUSSIOtT OP ¥= SIH Z.

Applying the tests stated in Art. 2, to V/ = sin z, v/e

have :

pi(x + iy) _ Q-iix-tiy) (1) u+ iVz -S -g-^-A

- e"-^ (cos X -f i sin x)- e^ (cos x - i sin xi

2i ' '"~

__ e^ + e~y

sin x + 1

. ey - e-y

Equating reals and imaginaries , we have,

cos X

ey -i- e-y

U- ~ — -^--^ — sm x;

cos X, ana

3 u , ey-f e-y a X ^ 2

cos x;

ay" 2

— z and the function is analytic. • ' 3x a y'

py + p-y

Since — — - hyberbolic cosine y, and

ey - e-y ^

2 — ' hyberbolic sine y, we can write, U = cosh y sin X

V — sinh y cos x

(2) The tv70 systems of curves,

IJ =^ cosh y sin x

V =: sinh y cos x

are orthogonal, or, stated otherv/ise, U and V are conjugate functions of x and y.

(3) The mapping of the Z-plane into the V/-plane is a con- formal representation.

(4) Since cosh y is a single-valued and continuous function for all values of y within the limits, 0 = y^"^ , and sin x is a single-valued and continuous function for all values of x in the interval, i^RJlJjJL ( x<i^-±™~-* integer).-

6

(2n + l)jr < < (2n 5)7T 2 - ^ = ' 2

Ther t?f ore ,

U - cosh y sin x, is a single- valued and continuous function of U v;ithin the region, 0 < y ,

Similarly, since sinh y is a single-valued and continu- ous function for all values of y and cos x is a single-valued and continuous function for all values of x within the interval, n7r i X i (n -f ijTT, therefore,

V — sinh y cos x, is a single valued and continuous function of V within the region _ < y< , nTri X ^ (n + ^y^'

Following directly from the above,

¥ = U -h iV,

is a single-valued and continuous function of x,y v/ithin the cormnon region. That is, ¥ is a single-valued and continuous function of both x and y within the region,

0^ y <^ , n^^x4(n-t- .

(5) To determine the period o|* sin z, let

sin z =: sin (z +c7C ) ^ v/here = a H- ib .

Solving and equa.ting reals and imaginaries, as follows:

ei(z + cK ) _ g-i(z -(-^ )

e_if_-_ ej^i2 _

2i '

2i ■

e'^Ccos x + i sin 2lLtJ??159A.JL 1 i sin x)

2i ^' '

,i(x+a)- (y-l-b) .^-i {x-fa)4 (y-i-b)

2i

e" |cos(x4 a ) 4-i s in ( x4-a^ " Q."^ "^^ -^c o s ( x+ a ) - i s in ( x-f a )}

(e-y-ey) ^ (e-y-t-ey)

2i

cos X -t-

2i

-cos (x-s-a)

sm X

- (y-fb). ,,y-tb

sin(x-f-a)

7

sinh y cos x — sinh (y^-b) oos(x-f-a)

cosh, y sin x cosh (y-f-b) sin(xta)

s inh J _ cos (x-f-a)

sinhCbty ) ~ cos x

cosh 2L_ _ sin(x-(-a) coshrb-tyT ~ sin X

Since the hyperbolic functions are not periodic, these equations

can only hold when b — 0, and sin(x-ha) — sin x & cos(xta) — cos x.

That is to say, a— 2tk ( k an integer) , and the period of sin z

equals th period of sin x.

Art .4.- MAPPI^ICt OF SI2T Z.

(l) ZP^42i^^il'^JLi refrion .

Le^ us choose a region in the Z-plane v/ith the following-

boundary:

and map this boundary of z and a few intermediate points, by means of the equations for U and V derived in Art.l.

U cosh y sin X (l)

V =, sinh y cos X (2) Consider a point in the Z-plane (see Pig. IV) such as a, v/here X is a constant, say equal to - ^-f-£ , ^ being an arbitrarily small number and y is negative, say equsil to -k. Substituting these values in equation (l), we have,

U ^ cosh(-k) sin{-^+e)

= (+K)(-l-fH) < -1 (cosh(-k)= K, a positive number considerably > 1 if |k/ is appreciable 0, and sin(-^ + e-)=- (-1+T|), >V being a small number <6 . ),

or, simply,

U <( -1.

Substituting in equation (2), we have,

8

V= sinh(-k) cos

= (-K') (n) - (sinh(-k)- -K', and cos (

or, simply,

V = - <^

These values, -1, and V n -S"^ are represented in the

W-plane by a point sur;h as A (see Fig. IV). Similarly points b,

c_, etc. map into points B, _C, etc., as shovm in the accompanying table and fif^ure.

: Let x= '

Let y =

Then U-

5.nd V— : Represented on : drawings by '.points such as : ( (Z-Plane) :¥-Plane:

-/-

<-/

. a ^

: b

A_ :

: B :

- £

: c

C :

', It

II

d

D :

e

¥ :

_-he^

f

F :

1

K

Cr :

: h

-1- -1- •

: <^

o

i

I :

: 0

o

.i :

.T

FIGURE IV.

Q rj

o o a c oJ / /% / / /

-I

> ' i' '"J

x^/ / / /

B D

o

E3

a

h X

/ / / /x't'/ / f

\ ////////, y ////// X p ///////^// 77- v's

9

This mapping shows that the ro.ci-ion in i he Z-plane, bound- ed as follows;

# -J^ xif •. Ol y

-^<x<^; 0>y, is a fundamental region, that is, maps into the entire W-plane. (2) Relation between the four quadrants .

let us consider soine point, say, a ib, in the first . quadrant of the indicated fundamental re,<^ion of the Z-plane . We have,

:z =■ a+ib:U-cosh b sin a = A:V=Lsinh b cos a = B:W=:.A-^iB :

Similarly for the other quadrants.

:z--a. t ib

U=cosh b__sjLnJj^a)5-A_

.V=sinh b cos(-a)= 3

.\V--A+ iB

:z=-a- ib

U=:cosh-b)sin(-a)r;-A

, V=:sinh-b ) cos ( - a):r-B

'w--A.- iB

:z- a-ib

U- cosh( -b ) sin a =;A

:V— sinh(-b) cos a--B

W= A- iB •

That is, the sine of any complex number, -a-f-ib, in the second quadrant, is equal to the sine of a -^ ib in the first quadrant, with the sirrn befor^ th^ real element c-ia.nc^ed to minus. The sine of any comiilex number, -a - ib, in the third quadrant, is equal to the si^-n of a-f-ib v/ith the si ^ns of both the imaginary an.i the real elements changed to minus. The sine of any complex number, a - ib, in the fourth quadrant, is equal to the sign of p + ib with the sign of the imaginary element changed to minus.

These simple relations give us a method for determining the sines of complex numbers in the second, third, and fourth quad- rants i:^ thos'j of th' first are known. It will therefore be

If There two regions together comprise the fundamental region. The equality signs are left of r of the limits for x in the one case because the entire line, x = ^ , v.'ould map tv/ice into the U- axis from 1 tocj^ . Likewise, but half of the line >: is used,

in order that it shall map but once into the U-axis from -1 to-^.

10

sufficient to map the f ir -t quadrant only.

If the lines x-= c (ge a const rjit), and. y— c, in the Z- plane, are mapped into the ¥-plane, the value of the sine of any oomplex number, a -fib, may be found by reading the values of U and V in th^: V -plane, v;here the mappin'-':s of x= a and y— b intersect. ^ ^ ^ ''Tiappin':^ of lines paral lei to_ the axes .

To map the lines, x— c, we eliminate y from equations, U = cosh y sin x sinh y cos x,

and to map the lines, y=c, v;e eliminate x from the same equations, as follows:

U— cosh y sin x ^ c 0 slT^y '

. 2

Addiig, v/e have,

U2 , _ V2 _ ^ ^ cosh^ sinh^

V— sinh y cos x -9— r cos^x

sinh y

(1)

Attain, we f^et,

■v-^~ — cosh^yj sm^x

cos"^

-— sinh^y

Subtracting, we have,

r2

sm^x

V

cos~x

— =: 1

(2)

tion;

Plotted upon rectangular coordinates the following equa-

■j^2 ■^»2

represents an ellipse, intercepting the axes at x = i a, and y=± b, and with foci at iVa^-b? Equation (l) is of this type, in

! 11

which cosh y= a, Binh y=h, U~x, and V— y. Therefore, if y is a parameter actuation (l) reijresents a system of ellipses, intercept- ij ing the IJ and V axes at ± cosh y, and d: sinh y respectively, and having foci at i 1, since ±\/ cosi-py - sinTr^y~=^l. Again, the equation,

j reijresents an h;>'-perbola, intercepting'- the x-axis at x = J: a, and havin?^ foci at dt \/a/^-f~b^ . Kqua.tion {2) is of this type, in

i which sin x=: a, cos x=b, U=x, and V = y, Therefore, if x is a parameter, equation {2) represents a system of hyprrbolas, inter- cepting the U-axis at ± sin x and having foci at ± 1, since ± V sin<^x -f- cos^x -± 1 .

ij In equation (l), as y takes in turn, larger and larger

I const c?Jit ■ values , cosh y and sinh y take ever increasing values

j and the intersections of the ellipses vrith the axes become more

I and more distant from the origin.

In equation (2), as x teJ-ces in turn lar.Ter and larger

' constant values, always <^ , sin x increases up to 1 as a limit, that is, the intersections of the hyperbolas with the U axis travel

i

from 0 to 1.

i The nature of this mapping for the first quadrant is

shown in map I.

i

Art. 5.- EXPLAiJATIOlT 0? THE JAPS.

( 1 ) '"General description of all the maps .

The cross-ruled paper upon V7hich the mapping is made represents tho first quadrant of the "Vv-pls,ne . As explained in : Art. 4, the snapping for one quadrant is sufficient to determine

the values for the entire plane. Por values of U, distances a- ' long the horizontal axis are measured, and for values of V, distanc- es along the vertical axis. The numbering in green ink indicates ; the values of U and V. The ellipses are the mappings of the lines in the Z-plane, a constant. The value of this constant

is indicated by the numbering in black ink along the left and part of the lower margin. The first ma,p is constructed for values of y as follows:

! 0 4 y 1 2.30

The second for,

I 2.30^ y4 3.90

The third for,

3.90 4 y i 5-50 Art .6 gives a m thod for finding the values of the sines of complex numbers in which the imaginary element is > 5.50.

The hyperbolas are the mappings of the lines in the Z-plane, x = a constant. The value of this constant is indicat- ed by th:3 numbering in black ink nenr the origin and along the boundary most distant from the origin. In all the maps, ' 0 < X 1 f . 1 (2) Ma£ I r ( 0 4 y ^ 2.50 ) I The scale in this map is, 1 211m. — .02, for values of ^ U and V. It is to be noticed that the mapping grows dense as !j we approach the point, U=l, V — 0 , and for this reason fev/er el- lipses and hyperbolas are plotted in the region near this point than in the more distant regions. This is true of all the maps. 1 The following outline gives the difference in the values of y for adjacent ellipses that are mai^ped, and the difference in the !

13

value of X for adjacent hyperbolas that are mapped in the various

reactions of map It Similar outlines will be found for maps II

and III.

Por ref^ion' bounded : as follows:

--J.>x > 1 ) Adjacent ellipses differ by .1 . jA^^JL ) Ad j ac ent hy :-er bo las differ by .05.

1 > X > .5 ) Adjacent ellipses differ by .05

^ > X-> 0 I and

t Jl X .5l_*A - /Adjacent hyperbolas differ by .05

.5 > X > 0 lAdjacent ellipses differ by .05 .5 > y ;> 0 ) Adjacent hyperbolas differ by .02

In the remaining )Adjacent ellipses differ by .02 region )Adjacent hyperbolas differ by .02

(3) £^a£ 11. (2. 50 4 y 4 5.90)

The scale in this map for values of U and V is 1 mm.= .IJ

For region boimded : as follows:

2.50<y<5.00 ) Adjacent ellipses differ by .05 „_ jAdjacent hyperbolas differ by .05

5.00<y<5.90 )Adjacent ellipses differ by .02 ) Adjacent hyperbolas differ by .02

ITote that the ellipses are practically circles and the hyperbolas are nearly straight lines in this map. lap. LLl' (5.90 I y = 5.50)

The scale in this map for values of U and V is 1 mm. .5.

For region bounded as follows:

5.90<;y<4.50 )Adjacent ellipses differ by .05 )Adjacent hyperbolas differ by .02

4.50<y<5.50 )Adjacent ellipses differ by .02

) Adjacent hyperbolas differ by .02

In this map the ellipses, as constructed, are no longer

ellipses, but circles, having their center at the origin, and the hyperbolas, as drawn are straight lines, intersecting, if pro- duced, at the origin. (^^^ee Art. 6 (a)) "Hescription of ellipsograph .

The ellipses in the maps were drawn with the help of an ellipsograph. The ellipsograph used consists of tv70 main parts, the frame and th( bea^n compass. The former forms a "T", the arms and stem of which are to lie along the major and minor axes, respectively. In both arms and ste i are slots, in which clamps, bearing the beatii compass, are free to slide. To set the instrument to dra\v a given ellipse the clamp running in the slot along the arms is claraped to the beam of the beam compass so that the distance between the pen point and the clarap is equal to one- half the minor axis of the desired ellipse. Similarly, the distance betv/een the other clamp and the pen point is made equal to one-half the major axis. . Under these conditions the pen point will trace the desired ellipse, because of the fact that for a given ellipse, if any line cuts the circumference at right angles the distance intercepted by the axes is constant.

Accompanying is a cut showing the instrument, taken from the catalog of Eugene Dietzgen Company.

Pin UKE V

15

(6) Construction of the maps «

In the construction of the ellipses all the data neces- sary was the values of the laajor and minor axes. Prom the equation,

"cosh^ ^ silih^ ~ ^ we see that the intersection of the ellipse v/ith the U-axis (or major axis) is at the point U - cosh y, and that the intersection with the V-axis (or minor axis) is at the point V sinh y. Cosh y and sinh y were obtained from a table of hyperbolic func- tions#. The ellipses for the three maps v/ere drawn in this manner, but it is to be noticed that for the greater part of the second map and for all of the third the difference between cosh y and sinh y is less than .05, or, expressed in millimeters, less than .5 mm. in map II and less than .1 in map III. In the case of map III no attempt v/as made to express this sli/rht dif- ference.

The hyperbolas were constructed by drawing a smooth curve through a certain number of calculated points. For map I the intersections of the various hyperbolas with the ellipses y - 0 (this is simply the U axis from 0 to l), y-.50, y-^1.00, y= 1.50, and y=2.25 were calculated from the following equations:

IT = cosh y sin x

V — sinh y cos x For map II the intersections of the hyperbolas with the ellip- ses, y- 2.30, y =. 3.00 and y— 3.90 were sufficient. raa.^

# Tor tables consulted see Bibliography.

Ill the intersections of the hyperbolas with the single ellipse y— 5.50 were all that were needed, because the system of hyper- bolas is orthogonal to the system of ellipses and the ellipses in this map are not distinguishable from circles, therefore the orthofronal rays, if produced, ?;ould pass through the origin, so the origin is available as a second point in determining the hy- perbolas. (See Art .6 (a) ).

Art .6 .- EVALUATION OP SII£__a±_ib, V^TIERE b> 5.5.

The accuracy with which values may be read in any of the maps is less than 7^ mm. This corresponds to a value of .125 in U and V in map III and to a value of .025 in U a^nd V in map II. The values of U and V are;

U = cosh Y sin x

V = sinh V cos x, or , IJ - g sm X

V - - — ~- — cos X

IT - sin X -f ~- sin x

V.' — ^ COS X - -^-Tr— CO s X

2

If y is so large that,

sin X < .025

COS X < .025,

then U and V may be written,

sin X

17

v/ithout introducinpc an error which affects the graph to a notice- able decree. When y 3 .

sin ^ - .024895

cos 0 .024895

Therefore the following relations hold for our purposes: (a) U , V _ ,

WW

This is the form which the equation of the ellipses takes when p- y

~— IS SO small that we can neglect it. It is the equation of a circle with radius ^L- and center at the origin. The ortho-

gonal system has this equation;

U2 Vg _ ey _ ^

— ■ — ^ — — — ~ — — 0

sm'^x cos'^x 2 2 —

U - (tan x) V,

which is the equation of a system of straight lines intersecting at the origin.

(b) Given, U + iV = sin (x+iy). If y > 3

TT_ ey .

U— ^ sm X 2

V- ^ cos X

10 (U + iV) =1 sin (x'-f iy' ) lOUzr sin x'

lOY- ^ — cos x'

U-- e^ ' sin x' — e^ sin x 20 ~~ 2 "

V — ^ e^ ' cos X ' e^ cos x 20 " 2

18

Dividing: , we have,

sm x' _ sm X

Therefore ,

tan x' tan x x' rr X

ey' _ ey

20-2

10 - ^ ey'-= lOey = ey-^2.3026-

y' y-h 2.3026-

and, , 10 (U-hiV) = sin [x-f i (y-f- 2 . 3026- )] =. 10 sin (x-f iy) (03) That is, the sines of numbers, a-|- ib, where 5.5, the largest value for v/hich the maps are constructed, may be obtained by findin*? the value of sin[af i(b - 2.3026 )j , and multiplying by 10. If (b - 2.3026)^ 5.5, look up the sine of ^-fi(b -n2.3026)] and multiply by 10^. n is a positive integer and must be chosen so that 3. i (b - n 2.3026) 4 5.5.

'77' 77'

Art .7 .- Evaluation of sin a -t ib, v/here i_^_^_aj_J?£.^j^» From Art. 3 (5) we have the following relation: sin (a-l-ib) = sin (277k+a+ib) U= cosh b sin a=^cosh b sin(a + 21<:77) = cosh b(-l)^sin (a+k7r) V-=L sinh b cos a=: sinh b cos { a+ 2k7r) = sinh b( -l) ^cos (a+k?/), where k is a positive or negative integer, so determined that - ^^(a4- kT^J. Therefore, considering absolute values only, of U and V,

|u! + i |v| = sin(a -h ib)

-f- i (v| - sin(a-f- kTT-fib) The sign of U is the same as the sign of (-l)^sin(a+ k?/) , and

J

19

the sign of V is (-l)''^, when b is positive, in the equation U+iV — sin a -Mb. Or, in general, the sign of U is the same as the sign of ( -1 ) ^sin( a-+- k?ir) , and the sign of V is the same as the sign of (-l)^b. By this means we can relate the sine of any complex number with some other complex number lying within - and for the real part.

In Pxt . 4 (2) the relation between the four quadrants is given, so we can nov^/ express the sine, of any complex number in tfrms of the sine of some complex number in the first quadrant. This relation is expressed by the following eq.uations, in v/hich a and b are axiy real numbers, and k an integer so determined that (a + kTrX ^.

sin( a -t- ib ) ^ U + iV sin( |a -^ kTTl + i |bl ) = u + iv

Tja-f- k?r|)

( V= (-1)^ V

This relation, together with,

B ( sin(a-rib)^ 10 sin[a + i(b - n 2.302585)] , needed only when b > 5.5, enables us to find the sine of any complex number to the degree of accuracy for which the maps are constructed .

The follovifing drav/ing shows the regions in the Z-plane mapping into the same re-^^ions in the W-plane as the four quadrants in our chosen fundamental region;

20

PirtiiKR VI

X

zn:

HI

r

ZE5Z

XT

X

-STr

-ZTT

rr

sv

^3z:

XT

35^

X

I25Z:

Art ^ ACCURACY OF Tiff: £IAPS .

The precision with which the maps may be read accurate- ly is practically to the nearest rmii. The errors of construc- tion probably do not exceed .3 ram. and v/ith careful reading an error greater than .3 mm. should not be made on this account. Therefore in map I the values of U and V are correct to ± .01, in map II to ±.05, and. in map III to±.3. The percentage error in the value of the modulus varies in all the maps inversely as the distance from the orir:in, being, approximately, one percent at a distance of 5 cm. and .2 percent at a distance of 25 cm. In the following article the actual error and the percentage error is shown for several examples.

The percentage error in the modulus is calculated by finding the modulus, obtained from the values of U and V as giv- en by the maps, and comparing with the true modulus, calculated from the values of U and V from the equations,

U — cosh y sin x

V =z sinh y cos X

The calculation of the percentage error in a single case, having

21

given tables of the hyperbolic functions and the ordinary trigono- metric functions, is somewhat of a task, involving 12 operations,

as follows: ( 2 multiplications

4 squarings 2 additions

2 extractions of square root! 1 subtraction 1 division

Art. 9.- EXAIvDPLES.

Let it be desired to find the sine of (1.320-f-i 2.024). Turn to map I. Follow along either the U or V-axis until 2.00 is reached. The next ellipse is 2.02 and the next 2.04. Esti- mate £^ of the distance betv/een 2.02 and 2.04 and follow the ellipse to the point where it intersects v;ith the hyperbola. 1.32, found most readily by referring to the right boundary of the map. Keeping this point of intersection follow down the green lines to the lower boundary and determine the value of U by reference to the numbering in green ink. This value is found to be ap- proximately 3.731. Erom the same point of intersection follow the green lines to the left or right boundary and determine the value of V by reference to the numbering in green ink along these boundaries. This value is found to be approximately .922. This gives,

sin (1.320-fi 2.024)= 3.73H-i .922

The correct value is,

sin (1.320 + i 2.024) =^ 3.730 + i .923,

showing an error in the modulus of .0008, or .02 per cent.

The examples in the following tables are chosen at

random::

22

:Sine of the

Values from maps *

Correct

values

error?

: number'

: U

: V

U

V

in ;!

:moduli_ -

: .721fi .362

.702

.270

. .70386-

.2779

: .6

Map li '

. .445-f- i2.241

2.050

'. 4.201

. 2.0469

4.2155

.5

y <,2.3:

.7504-1 .750 ;

.894

.594 :

.8825 '

.6017

' .6

1.320 4-12.024 ■

3.731 '

.922

3.7299 :

.9230

', .02

■ ' ■ ' ' ' .252-1-12.860 ;

2.17

8 .41

2.184 ;

.8427 '

.2

Map II

2.3<7^.9

.755-fi3.427 :

'.10.57 :

11.28 :

10.559 :

-.11.198

: .4

1.444 4-13.860 •

23.56 :

iJ!..'2^ J

_23^.553__J

. ,3 . 001

:: .04

.790-}- 14.463

30.75 :

30.60 :

:30.817 :

30.522

: i . .02 1

Map III'

>.9?<y<5.5

1.2454-14.988

69.65 •

23.54

69.468 :

23.469

: . 5) '

\ .3624-15.456

41.32 :

108.90 •

41.466 ■

109.503

.6

See foot- note

Prin- ciple used

To obtain sine of number below

Look up sine of this number

Values from maps

u

A and B

1.06 -f- 16.104 :1. 06 + 13. 801 sin(l .06 -f 16 .104 ) - 195 .6 4 iiP,^_«9

.53-1- 17 .593 : .53-1- 15.290 sln( .53-f- 17 .593)= 502.5 -M857.2

2.242-M2.024 : .900+ 12.024

3.500+ 14.00 : .358+ 14.00 injj5^. 50 0 + 1 po]p_ -9 -_ i 2j;315_ 5.8024 128.000: .4814-14.974

19 .56

50 .25

3.020

10.99

85 .72

2.315

9 .50

33.50

sm

(5.802+ 128.)^ 10^° (-33. 50 4 164.)

A and B

7 . + ilO.

.717 -+ 15 .39 5;^

:72.42

25.50

64.00

83.00

sin (7 4- 110)= 7242.-/- 18300.

23

(Table Continued)

See foot- note

8

Prin- ciple used

A and

B

To obtiaim sine of nu:nber belov/

-4 . + i4.42

Look up sine of this nuraber

.85B + i4 .42

;in(-4 . + i4.42)- 31.'49 - i27.37

-17.908 -i34.446: .942+i4.512

Values from maps

u

31.49

36 .67

27 .37

26 .85^

sin(-17 .908 - i34 .446 ) = 10^"^ ( 36 . 67 - i26.85)

sm

sm

sm sin

sin sin

sm sin

1.06 -f i6. 104)= 10lsin(i.06 + i(6 .104 -(1)2.303)]

= 10 sin(1.06 -|-i3.80l)

.53-fi7 .593)-10^sin [.53+ i(7 .593 -(1)2.303)}

=10 sin( .53 4-i5 .290) ____

2 .242 -f i2.024--U-h iV

1 2.241+ (-l)7r( + i2.024)-u+iv

-1)-1 i2^4_2^-r) ^ (2.242 -Tr(

^' 12.0241

3 .5 i i4.) ^ U + iV

I 3.5+ (-l)7r| + i4. ) u+iv _^)-l (3.5 -7/) _

-1)

-1 ± H

-V

5.802+i28.) = U+iV

I 5.802 + (-2)'7r| + i[28.-(10)2.3026i )

= sin(|- .48l|-f i4.974)=u+iv 10)^°(-1)- -JLl^l 1 u--10,000,000,000u

10)1Q(-1)2 _^|j|74_ ^ ^ 10,000,000,000w

sin{7 . -MIO. ) -U + iV

sin(l7. + (-2)7r| + i[lO -(2)2.3026] )

^ sin( .717 -f i5.395) = u -tiY U-^(10)^(-1)"^ -j^f^ u ^ lOOu

V- (10)2(-1)'^ ^^j-- V - lOOv

sin(-4. -f-i4.42) = Uf-iV sin(|-4. + (l)7r| -f i4.42)

= sin( I - .858 1 -f i4.42) = u -^-iv

/ -, \1 4 .42

V ::.(-!) —^427 -V

sin(-17 .908 - i34.446)^ U-f iV

sin( j-17.908 4-(6)Tr| -f i|-[34.446 -(13)2.30259] )

=^ sin( |.942| f i|-4.512| ) =^ u + iv u- (lO)-'-'^(-i)^ "iTtlr ^ - 10,000,000,000,000u V3= (10 ^^(-1)^ _^||l|46_ v^-10,000,000,000,000v

For convenience of reference the follov/ing tables, giving multiples of TT" and of 2.3035850 up to 20, are subjoined

Multiplier

X- rr :

'Multipliei

' X logelO

: 1 ;

3.14 1 59265:

1 '

2.30258509

: 2 :

6.283 ;

2

4.605

: 3

9.425 "

3

. 6.908

: 4

12.566 •

4

: 9.210

: 5 :

15.708 '

5

11.513

: 6 :

18.850 :

6

• 13.816

: 7 :

21.991 :

7

. 16.118

: 8 :

25.133 :

8

18.421

: 9 :

28.274 :

9 ;

20.723

: 10 :

31.416 '

10

. 23.026

: 11 :

34.558 :

11

25.328

: 12 ;

37.699 :

12

27 .631

: 13 :

40.841 '

13

29.934

i.4

, O t ,y » <C O D

: 15 '

47.124 •

: 15

: 34.539

: 16

50.265

: 16

: 36.841

: 17

: 53.407

: 17

: 39,144

: 18

: 56.549

: 18

• 41.447

: 19

: 59.690

: 19

• 43.749

: 20

: 62.832

: 20

: 46.052

26

Art .10.- EVALUATIOIT OP COS z, TAIT z, COT z, SEC z, AlH) CSC z. (1) Cos z.

W= U -t-iV=^cos z^ cos(x + iy) — — J— ^

_ e-y(cos x + i sin x) - ey(cos x + i sin x)

™^ COS x+i g. ---^ sin X

= sinh y cos x + i cosh y sin x U sinh y cos x, V — cosh y sin x

Observing that in, sin(x + iy),

V — sinh y cos x, U — cosh y sin x, we see that

all we have to do to find the cosine of a complex number is to look up the sine and interchange the real and imaginary elements, Tan z, coJ'^z, sec z, esc z.

Having sin z and cos z the ordinary trigonometric rela- tions give us the other functions:

. sin z

, cos z

cot Trrrr" sm z

sec z

cos z 1

2 ^ sm z

27

BIBLIOGRAPIIY"

An extended search revealed nothing upon this particu- lar subject, and the following books, containing tables and dis- cussions of the hyperbolic trigonometric functions were all that were used:

Ligov/ski, W,, Tafeln der Hyperbelf unct ionen und der

Kreisf unctionen, Berlin, Verlag von Ernst &

Korn, 1889. Burrau, Carl, Tafeln der Punkt ionen Cosiaus und Sinus,

Berlin, Verlag von Creorg Keimer, 1907 .

Blakesley, Thomas H., A table of hyperbolic cosines

and sines, London, Taylor and Francis,

1890.

I

P I IT I S